Question

An electromagnetic wave travels in vaccum represented by, $$ \overrightarrow{\mathrm{E}}=\left(\mathrm{E}_{1} \hat{i}+\mathrm{E}_{2} \hat{\mathrm{j}}\right) \cos (\mathrm{kz}-\omega \mathrm{t}) $$ then select incorrect option: (1) wave travels along positive \(z\)-direction (2) the associated magnetic field is \(\overrightarrow{\mathrm{B}}=\frac{1}{\mathrm{c}}\left(\mathrm{E}_{1} \hat{\mathrm{i}}+\mathrm{E}_{2} \hat{\mathrm{j}}\right) \cos (\mathrm{kz}-\omega \mathrm{t})\) (3) The associated magnetic field is \(\overrightarrow{\mathrm{B}}=\frac{1}{\mathrm{c}}\left(\mathrm{E}_{1} \hat{\mathrm{i}}-\mathrm{E}_{2} \hat{\mathrm{j}}\right) \cos (\mathrm{kz}-\omega \mathrm{t})\) (4) The given electromagnetic wave is plane polarized light

Short answer

Option 2 is incorrect.

Step-by-step solution

- Identify the Wave Direction

The wave equation \[ \overrightarrow{\mathrm{E}}=\left(\mathrm{E}_{1} \hat{i}+\mathrm{E}_{2} \hat{\mathrm{j}}\right) \cos (\mathrm{kz}-\omega \mathrm{t}) \] indicates that the wave is traveling in the positive z-direction because the argument of the cosine function is (kz - ωt). This means the wave progression is in the +z direction.

- Examine the Magnetic Field Options

Using Maxwell's equations, the magnetic field component of an electromagnetic wave traveling in vacuum can be derived. The E and B fields are perpendicular to each other and to the direction of propagation.

- Analyze the Magnetic Field Options (Option 2)

For the magnetic field given in option 2 \[\overrightarrow{\mathrm{B}}=\frac{1}{\mathrm{c}}\left(\mathrm{E}_{1} \hat{\mathrm{i}}+\mathrm{E}_{2} \hat{\mathrm{j}}\right) \cos (\mathrm{kz}-\omega \mathrm{t}) \] This would mean the magnetic field is in the same direction as the electric field. But according to the right-hand rule and Maxwell's equations, this is not possible as the magnetic field should be perpendicular to the electric field.

- Analyze the Magnetic Field Options (Option 3)

For the magnetic field given in option 3 \[\overrightarrow{\mathrm{B}}=\frac{1}{\mathrm{c}}\left(\mathrm{E}_{1} \hat{\mathrm{i}}-\mathrm{E}_{2} \hat{\mathrm{j}}\right) \cos (\mathrm{kz}-\omega \mathrm{t}) \] This magnetic field configuration is possible because it is perpendicular to the electric field configuration containing (\mathrm{E}_{1} \hat{\mathrm{i}}+\mathrm{E}_{2} \hat{\mathrm{j}}). Therefore, this option is correct.

- Plane Polarized Light

The electric field has components only in the x and y directions, meaning it lies in the xy-plane. Hence, the given electromagnetic wave is indeed plane polarized light.

- Determine the Incorrect Option

Based on the analysis, option 2 is the incorrect one because it suggests a magnetic field that is not perpendicular to the electric field.

Key concepts

Electromagnetic Field Theory

Electromagnetic Field Theory explains the behavior of electric and magnetic fields and their interactions. It is grounded in Maxwell's equations, which unify electricity and magnetism into a single theory.

In this theory, the electric field (E) and magnetic field (B) vectors are perpendicular to each other and to the direction of wave propagation. They oscillate sinusoidally and travel through space as an electromagnetic wave.

For our problem, the electric field is given as:

\[\frac{{\text{\overrightarrow{\text{E}}}}}{{\text{E}_1 \textbf{\hat{i}} + \text{E}_2 \textbf{\hat{j}}}} \cos (\text{kz} - \text{\omega t})\]

This represents an electromagnetic wave traveling in the z-direction with components of the electric field in the x and y-directions.

Maxwell's Equations

Maxwell's Equations are four fundamental equations that describe how electric and magnetic fields interact and propagate. They are:

\[ \text{abla \cdot E} = \frac{\rho}{\varepsilon_0} \] (Gauss's Law)
\[ \text{abla \cdot B} = 0 \] (Gauss's Law for Magnetism)
\[ \text{abla \times E} = - \frac{\partial{\text{B}}}{\partial t} \] (Faraday's Law of Induction)
\[ \text{abla \times B} = \frac{1}{c^2} \frac{\partial E}{\partial T} + \mu_0 J \] (Ampere's Law)

Using these equations, we can derive the relationship between electric and magnetic fields in an electromagnetic wave. In our example,

\text{B} \propto \frac{1}{\text{c}}\text{E}, where \text{c} is the speed of light. This relationship shows that the fields are proportional and perpendicular to each other.

Wave Propagation

Wave propagation describes how a wave travels through space. For electromagnetic waves, the direction of the wave is determined by the direction in which the phase of the wave propagates.

In the given exercise, we have:

\[ {E} \text{\cos} (kz - \omega t) \],
which indicates the wave travels in the +z direction.

The wave involves oscillating electric and magnetic fields, where both fields are perpendicular to the direction of propagation and to each other. This characteristic is crucial for understanding how electromagnetic waves move through different mediums and interact with materials.

Plane Polarized Light

Plane polarized light is a specific type of electromagnetic wave in which the electric field oscillates in only one plane.

In our scenario, the electric field has components in the x and y directions, restricted to the xy-plane. Hence, this specific wave is described as plane polarized.

This means, the electric field vector lies within a single plane (the xy-plane) as it travels in the z direction. Such polarization plays a key role in various optical technologies and is crucial for understanding how light interacts with materials like polarizing filters or birefringent substances.